WB JEE 2020

If the tangent to the curve y2 = x3 at (m2, m3) is also a normal to the curve at (m2, m3), then the value of mM is

In open interval \(\left( {0,\,{\pi \over 2}} \right)\)

Consider the curve \(y = b{e^{ - x/a}}\), where a and b are non-zero real numbers. Then

If the function \(f(x) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1\) [a > 0] attains its maximum and minimum at p and q respectively such that p2 = q, then a is equal to

A particle is projected vertically upwards. If it has to stay above the ground for 12 sec, then

Tangent is drawn at any point P(x, y) on a curve, which passes through (1, 1). The tangent cuts X-axis and Y-axis at A and B respectively. If AP : BP = 3 : 1, then

Let \(f(x) = {x^{13}} + {x^{11}} + {x^9} + {x^7} + {x^5} + {x^3} + x + 12\).Then

Area in the first quadrant between the ellipses x2 + 2y2 = a2 and 2x2 + y2 = a2 is

The area of the figure bounded by the parabola \(x = - 2{y^2},\,x = 1 - 3{y^2}\) is

The area of the region\(\{ (x,y):{x^2} + {y^2} \le 1 \le x + y\}\) is

If \({x^2} + {y^2} = {a^2}\), then \(\int\limits_0^a {\sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} dx = }\)

If c0, c1, c2, ......, c15 are the binomial coefficients in the expansion of (1 + x)15, then the value of \({{{c_1}} \over {{c_0}}} + 2{{{c_2}} \over {{c_1}}} + 3{{{c_3}} \over {{c_2}}} + ... + 15{{{c_{15}}} \over {{c_{14}}}}\) is

The equation of circle of radius \(\sqrt {17}\) unit, with centre on the positive side of X-axis and through the point (0, 1) is

The locus of the centre of the circles which touch both the circles x2 + y2 = a2 and x2 + y2 = 4ax externally is

The equation \(z\bar z + (2 - 3i)z + (2 + 3i)\bar z + 4 = 0\) represents a circle of radius

The number of complex numbers p such that \(\left| p \right| = 1\) and imaginary part of p4 is 0, is

The value of \(\sum\limits_{n = 1}^{10} {} \int\limits_{ - 2n - 1}^{ - 2n} {{{\sin }^{27}}} x\,dx + \sum\limits_{n = 1}^{10} {} \int\limits_{2n}^{2n + 1} {{{\sin }^{27}}} x\,dx\) is equal to

Let f, be a continuous function in [0, 1], then \(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 0}^n {{1 \over n}} f\left( {{j \over n}} \right)\) is

\(\int\limits_0^2 {[{x^2}]} \,dx\) is equal to

The differential equation of the family of curves y = ex (A cos x + B sin x) where, A, B are arbitrary constants is

If \(x\sin \left( {{y \over x}} \right)dy = \left[ {y\sin \left( {{y \over x}} \right) - x} \right]dx,\,x > 0\) and \(y(1) = {\pi \over 2}\), then the value of \(\cos \left( {{y \over x}} \right)\) is

Let cos\(^{ - 1}\left( {{y \over b}} \right) = \log {\left( {{x \over n}} \right)^n}\). Then

Let \(y = f(x) = 2{x^2} - 3x + 2\). The differential of y when x changes from 2 to 1.99 is

Let f be a differentiable function with \(\mathop {\lim }\limits_{x \to \infty } f(x) = 0.\) If \(y' + yf'(x) - f(x)f'(x) = 0\), \(\mathop {\lim }\limits_{x \to \infty } y(x) = 0\), then (where \(y \equiv {{dy} \over {dx}})\)

Let \(y = {1 \over {1 + x + lnx}}\), then

Let \(y = {{{x^2}} \over {{{(x + 1)}^2}(x + 2)}}\). Then \({{{d^2}y} \over {d{x^2}}}\) is

Consider a tangent to the ellipse \({{{x^2}} \over 2} + {{{y^2}} \over 1} = 1\) at any point. The locus of the mid-point of the portion intercepted between the axes is

If B and B' are the ends of minor axis and S and S' are the foci of the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over 9} = 1\), then the area of the rhombus SBS' B' will be

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Let \(f(x) = \sqrt {{x^2} - 3x + 2}\) and \(g(x) = \sqrt x\) be two given functions. If S be the domain of fog and T be the domain of gof, then

Let \(A = \{ x \in R: - 1 \le x \le 1\}\) and \(f:A \to A\) be a mapping defined by \(f(x) = x\left| x \right|\). Then f is

The domain of \(f(x) = \sqrt {\left( {{1 \over {\sqrt x }} - \sqrt {x + 1} } \right)}\) is

Let \(f(x) = 1 - \sqrt {({x^2})}\), where the square root is to be taken positive, then

A double ordinate PQ of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) is such that \(\Delta OPQ\) is equilateral, O being the centre of the hyperbola. Then the eccentricity e satisfies the relation

\(\int {{{f(x)\phi '(x) + \phi (x)f'(x)} \over {(f(x)\phi (x) + 1)\sqrt {f(x)\phi (x) - 1} }}dx = }\)

Let f : R \(\to\) R be twice continuously differentiable (or f" exists and is continuous) such that f(0) = f(1) = f'(0) = 0. Then

Let \(\phi (x) = f(x) + f(1 - x)\) and \(f(x) < 0\) in [0, 1], then

Let \(f(x) = {1 \over 3}x\sin x - (1 - \cos \,x)\). The smallest positive integer k such that \(\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^k}}} \ne 0\) is

Let \(0 < \alpha < \beta < 1\). Then, \(\mathop {\lim }\limits_{n \to \infty } \int\limits_{1/(k + \beta )}^{1/(k + \alpha )} {{{dx} \over {1 + x}}}\) is

\(\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)\)

If \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + cx} \over {1 - cx}}} \right)^{{1 \over x}}} = 4\), then \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 2cx} \over {1 - 2cx}}} \right)^{{1 \over x}}}\) is

The equation \({x^{{{(\log 3x)}^2}}} - {9 \over 2}\log 3\,x + 5 = 3\sqrt 3\) has

If 2 log(x + 1) \(-\) log(x2 \(-\) 1) = log 2, then x =

Let \(A = \left( {\matrix{ a & b \cr c & d \cr } } \right)\) be a 2 \(\times\) 2 real matrix with det A = 1. If the equation det (A \(-\) \(\lambda\)I2) = 0 has imaginary roots (I2 be the identity matrix of order 2), then

Let \(A = \left[ {\matrix{ {12} & {24} & 5 \cr x & 6 & 2 \cr { - 1} & { - 2} & 3 \cr } } \right]\). The value of x for which the matrix A is not invertible is

If \(\left| {\matrix{ {{a^2}} & {bc} & {{c^2} + ac} \cr {{a^2} + ab} & {{b^2}} & {ca} \cr {ab} & {{b^2} + bc} & {{c^2}} \cr } } \right| = k{a^2}{b^2}{c^2}\),then K =

If f : S \(\to\) R, where S is the set of all non-singular matrices of order 2 over R and \(f\left[ {\left( {\matrix{ a & b \cr c & d \cr } } \right)} \right] = ad - bc\), then

Let A = $$\left( {\matrix{
{3 - t} \cr
{ - 1} \cr
0 \cr

} \matrix{
{} \cr
{} \cr
{} \cr

} \,\matrix{
1 \cr
{3 - t} \cr
{ - 1} \cr

} \matrix{
{} \cr
{} \cr
{} \cr

} \matrix{...

If the vectors \(\alpha = \widehat i + a\widehat j + {a^2}\widehat k,\,\beta = \widehat i + b\widehat j + {b^2}\widehat k\) and \(\,\gamma = \widehat i + c\widehat j + {c^2}\widehat k\) are three non-coplanarvectors and $$\left| {\matrix...